Spirograph: The Mathematics of Rolling Circles
If you've ever played with a Spirograph toy, you know the satisfaction of creating intricate, looping patterns just by guiding a pen around a geared wheel. While it feels like magic or pure art, those mesmerizing curves are completely governed by mathematics.
These patterns are specifically known as roulettes—curves traced by a point attached to a curve as it rolls without slipping along another fixed curve. In the case of a standard Spirograph, the curves are circles, and the point is the tip of your pen.
Hypotrochoids: Rolling Inside
When a smaller circle (the "rotor") rolls around the inside of a larger fixed circle (the "stator"), the pen traces out a curve called a hypotrochoid.
The shape of the hypotrochoid depends on three things: the radius of the fixed outer circle ($R$), the radius of the rolling inner circle ($r$), and the distance of the pen from the center of the rolling circle ($d$). By changing the ratio of $R$ to $r$, you change the number of "petals" or loops the curve will have. If the ratio is a whole number, the pattern connects back on itself after just one trip around. If it's a fraction, the wheel has to go around many times before the line finally closes perfectly.
Epitrochoids: Rolling Outside
What happens if we put the gears on the outside? If the smaller circle rolls around the outside of the fixed circle, the curve traced is called an epitrochoid.
The math is nearly identical, but because the wheel is on the outside, the loops tend to point outward rather than inward. The same rules about ratios apply: the relationship between the fixed radius $R$ and the rolling radius $r$ dictates the symmetry and complexity of the final drawing.
Continuous Mathematical Curves
Whether you are drawing hypotrochoids or epitrochoids, the mathematical equations that describe these paths use sine and cosine functions, combining the circular motion of the wheel's center around the big circle with the spinning motion of the pen around the little wheel's center.
These equations are a perfect example of kinematics, the branch of physics that describes the motion of points, bodies, and systems of bodies without considering the forces that cause them to move.